Why is it important to follow the order of operations? What are some possible outcomes when the order of operations is ignored?

You should follow the order of operations because you need to get an accurate result. If these rules concerning order of operations would no exist, then each solver will solve an algebraic in a particular way and it is less probable to come to a correct answer.

Considering as an example the following algebraic expression which is solved in two ways: one hand,following the rules and, the other hand, disobeying the rules such that:

`4 + 5*x`

following...

You should follow the order of operations because you need to get an accurate result. If these rules concerning order of operations would no exist, then each solver will solve an algebraic in a particular way and it is less probable to come to a correct answer.

Considering as an example the following algebraic expression which is solved in two ways: one hand,following the rules and, the other hand, disobeying the rules such that:

`4 + 5*x`

following the rules: first multiplication, second addition: `4+5x`

disobeying the rules: operations performed from left to right, first addition, second multiplication:`(4+5)x = 9x`

Notice that disobeying the rules, the result is not accurate.

Hence, using PEMDAS(Paranthesis,Exponentiation,Multiplication,Division,Addition,Subtraction), you will come to the correct answer all the time.